While playing around with some simple melodies on my piano i realized that tapping (like a single tone in a trill) the tone one half-tone below in a melody before playing the actual tone sometimes sounds a lot 'more right' than just playing the tone alone.

4 Answers
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It's an approach note from below, and if it's a half-step below, it's a chromatic approach note. Why it sounds good:

(1) It makes the melody more varied and interesting, if not all notes start exactly the same. In your ear the short approach note and the target note after it blend together, so it can be seen as a different attack, different articulation.

(2) It can be seen as creating a short tension or a bit of dissonance that's resolved when the actual melody note arrives. Music is all about tension and release, consonance and dissonance. (both of which are subjective)

(3) Because the pitch moves ascending down-up, it creates more excitement and alertness than a descending movement, which is more soothing. (This is sometimes explained as being related to the doppler phenomenon where a rising sound means that someone or something - perhaps an enemy or a predator - is moving towards you, instead of away from you)

Edit: there's been some interesting discussion in the comments. I'll just add that these phenomena I listed are why I use the technique described by the OP. I use it to make note attacks less monotonic and therefore more interesting, to create short bits of tension/release by flirting with a "wrong" note, and to add some ascending motion if I think there's too much soothing descending motion in the melody. YMMV, but I feel these things very strongly, just like Dm6 - B7 - E7 - Am6 feels very different from Cmaj7 - D7 - G7 - C in a concrete and real way. You can easily try it yourself - does adding approach notes to a few melody notes make a boring melody feel more varied and interesting or not? Does descending pitch motion feel more soothing than ascending pitch motion?

Assuming a constant-pitch source, a rising pitch means that the source is accelerating toward you, not moving toward you. It could even be moving away from you, but decelerating.
– phoogApr 18 at 14:53

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Agreed with phoog's point. I think it may also be worth noting that acceleration here is meant in the vector sense (i.e. direction matters). If the source is moving closer to you but at an angle (e.g. a car passing by on the road while you're on the sidewalk), a continuous change in pitch may still take place, even if the car has constant speed. With that said, I'm not sure if the effect is significant enough to make a difference except at reasonably short distances.
– DanielApr 18 at 17:04

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@TracyCramer correct. But with a source at a constant frequency and a constant (relative) velocity, the target receives a signal that is also a constant frequency. The received frequency will be different from the emitted frequency because of the Doppler effect, but it will not vary. If the received frequency is varying, there must be a change either in the velocity or the frequency of the source.
– phoogApr 18 at 17:37

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@TracyCramer "what am I misunderstanding": if an approaching source resulted in a perceived rising pitch, then the sound of an approaching car or plane would rise until it passed you and then fall. But what you actually hear is that it begins at a constant pitch (which is higher than the true pitch heard by the occupants of the vehicle). It then begins to drop slowly (unless the source is heading directly toward you), drops more quickly as it passes you, and then drops more slowly until it stabilizes at a lower pitch and fades out.
– phoogApr 18 at 17:46

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@TracyCramer the frequency is indeed increased by the emitter and receiver moving toward one another. If the relative velocity is constant, then the increase in the frequency will also be constant, so you won't hear a rising pitch, but instead a constant pitch, which will be higher. A trumpet plays a B-flat a mile away: you hear a B-flat. A trumpet moving toward you at some velocity v plays a B-flat: you hear a C. For the pitch to rise from B-flat to C, the trumpet player must accelerate from rest to velocity v.
– phoogApr 20 at 4:58

This is a technique common in jazz, blues, and pop styles. It is essentially mimicking the way vocal, string, and wind players scoop or slide into a note. Since a true portamento is not possible on piano, it is simulated by quickly hitting a note above or below the target note slightly before the target note. Though today it is more common popular music, it is similar to embellishments we see in Baroque and Classical music. They are usually notated as grace notes.

@TracyCramer This piano technique sounds good because it mimics the human voice. A more detailed/scientific explanation of "why" would be outside the scope of this forum (which is concerned with theory & practice)
– PeterApr 18 at 18:13

The question doesn't limit the phenomenon to piano. Why do singers scoop? There are plenty of in-scope questions here on the subjects of psychoacoustics and other physical/scientific matters related to instruments, theory and performance.
– BeanlucApr 18 at 20:07

@Beanluc You will probably get a better response by posting this comment as a new question. The OP's question was about a specific piano playing technique, and I prefer to keep my answers short and factual ... a lot of the other comments and answers to this question seem very speculative to me.
– PeterApr 18 at 20:19

More generally, these are kinds of ornaments. As noted in another answer, this kind of ornament is also common in other styles of music.

The precise meaning of acciaccatura has changed over time. One definition holds that it should be struck at the same time as the principal note, but released immediately while the principal note is held. The other definition is that it should be played immediately before the principal note, as in the short appoggiatura.

The precise nature of the distinction between the short appoggiatura and the acciaccatura seems to be a matter of controversy. It will in any case depend on the period of a piece of music or even its composer.

Not covered in any of the other correct answers (although alluded to in the comments), this can also be a compensation at a subconscious level for the artificiality of the tempering process.

The natural scale is formed by mathematical relationships between the frequencies of pitches. But all natural scales are formed in relationship to a single root note, and are therefore "attached" to a single key. To make instruments more flexible and able to handle multiple keys and key changes, a technique called "tempering" was invented. This tweaks the notes of the natural scale to slightly different notes that sound almost the same. Now, instead of having a piano in the key of A or the key of C, you have a piano that handles all keys in the chromatic scale.

This flexibility, however, comes at a price. There's some reason to believe that people are subconsciously aware the intervals are just slightly off. For that reason, some musical styles, such as blues and jazz, use pitch bending, grace notes and accidentals to suggest notes closer to the natural scale tones. (If this interests you, I have an essay on the topic here.)